Gauge fields induced by curved spacetime

TL;DR

The paper addresses how gravity and gauge fields may be related in a lattice setting by revealing a triality among three lattice models: Dirac fermions in a periodic spacetime metric, nonrelativistic fermions in a gauge field (Harper–Hofstadter), and fermions in a periodic scalar field (Aubry–André). It shows these models are different representations of the same quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$ with $q^2= e^{i\omega}$, and that canonical transformations map between them, enforcing identical fractal spectra, topology, and transport properties. Key results include the mapping of the curved-spacetime lattice Hamiltonian to the Harper–Hofstadter and Aubry–André forms, the emergence of a Cantor Hofstadter-like spectrum in the incommensurate regime, and the quantized current associated with Chern numbers via topological pumping. The work suggests deep connections to dualities in gauge and gravity theories, offers concrete platforms for analog gravity experiments (e.g., Unruh-like effects in curved lattices), and provides potential new angles on quantum gravity in discretized spacetime. Overall, the triality extends S- and Langlands-duality concepts to spacetime curvatures and points toward novel quantum-gravity insights accessible through engineered lattice systems.

Abstract

I found an extended duality (triality) between Dirac fermions in periodic spacetime metrics, nonrelativistic fermions in gauge fields (e.g., Harper-Hofstadter model), and in periodic scalar fields on a lattice (e.g., Aubry-André model). This indicates an unexpected equivalence between spacetime metrics, gauge fields, and scalar fields on the lattice, which are understood as different physical representations of the same mathematical object, the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$. This quantum group is generated by the exponentiation of two canonical conjugate operators, namely a linear combination of position and momentum (periodic spacetime metrics), the two components of the gauge-invariant momentum (gauge fields), and position and momentum (periodic scalar fields). Hence, on a lattice, Dirac fermions in a periodic spacetime metric are equivalent to nonrelativistic fermions in a periodic scalar field after a proper canonical transformation. The three lattice Hamiltonians (periodic spacetime metric, Harper-Hofstadter, and Aubry-André) share the same properties, namely fractal phase diagrams, self-similarity, $S$-duality, topological invariants, flat bands, and topologically quantized current in the incommensurate regimes. In essence, this work unveils an unexpected link between gravity and gauge fields, opens new avenues for studying analog gravity, e.g., the Unruh effect and universe expansions/contractions, suggests the existence of an $S$-duality of spacetime curvatures, and hints at novel pathways to quantized gravity theories.

Figures (3)

• Figure 1: Triality between gauge fields, periodic scalar fields, and curved spacetime metrics on finite lattices. The Hamiltonian $\mathcal{H}_\text{AA}$ (left) of a nonrelativistic charged fermion on a 1D lattice in a periodic scalar field with frequency $\omega$ and phase shift $\phi$ (Aubry–André model) is equivalent under the transformation $(\omega \hat{x}, \hat{p}) \to (\hat{\pi}_x,\hat{\pi}_y)$ to the Hamiltonian $\mathcal{H}_\text{HH}$ (right) of a nonrelativistic charged fermion on a 2D square lattice in a gauge field with flux $\omega$ per unit cell (Harper-Hofstadter model). These two Hamiltonians are equivalent under a canonical transformation to the Hamiltonian $\mathcal{H}_\text{CS}$ (bottom) of a massless relativistic fermion in a periodic spacetime metric with frequency $\omega$. These three Hamiltonians $\mathcal{H}_\text{AA}$, $\mathcal{H}_\text{HH}$, and $\mathcal{H}_\text{CS}$ can all be written as $\mathcal{H}_{XY}=2\cos\hat{X}+2\cos\hat{Y}$ where $[\hat{X},\hat{Y}]=\mathrm{i}\omega$ and with the two canonical conjugate variables being respectively (left) $\hat{X},\hat{Y}=\omega \hat{x},\hat{p}$, (right) $\hat{X},\hat{Y}=\hat{\pi}_x,\hat{\pi}_y$, and (bottom) $\hat{X},\hat{Y}=\frac{1}{2}\omega \hat{x}+\hat{p},\frac{1}{2}\omega \hat{x}-\hat{p}$.
• Figure 2: Energy spectra of the Hamiltonian $\mathcal{H}_\text{CS}$ of a massless Dirac fermion in a periodic spacetime metric on a lattice of $N=210$ sites. (a) Energy spectra as a function of $p/q$ with periodic boundary conditions and $\phi=0$. (b) Energy spectra for $p/q=1/3$ as a function of the phase $\phi$ with open boundary conditions. (c) The metric $\alpha_n$ for $p/q=1/3$ and $\phi=0$ compared with $\cos{\left(\frac{1}{2}(\omega n+\phi)\right)}$ (dashed) as a function of the lattice site. (d) Wavefunctions of the two edge modes in the first band gap (from below) for $p/q=1/3$ and $\phi=0$. (e) Energy spectra for $\omega\approx2\pi(\Phi-1)$ (with $\Phi$ the golden ratio), as a function of the phase $\phi$ with periodic boundary conditions on a lattice of $N=233$ sites. (f) The metric $\alpha_n$ for $\omega\approx2\pi(\Phi-1)$ and $\phi=0$ as a function of the lattice site. (g) Wavefunctions of two bulk modes for $\omega\approx2\pi(\Phi-1)$ and $\phi=0$. (h) Energy spectra for small $\omega\approx0$, as a function of the phase $\phi$ with periodic boundary conditions on a lattice of $N=233$ sites. The spectra show several discrete levels at a short distance, which corresponds to a continuum of energy levels in the limit $\omega\to0$. (i) The metric $\alpha_n$ for $\omega=2\pi/233\approx0$ (taking $p/q=1/233$) and $\phi=0$ as a function of the lattice site. Notice that this metric becomes linear at the center $n=q/2$. (j) Wavefunctions of two bulk modes for $\omega\approx0$ and $\phi=0$.
• Figure 3: The approximation error $\delta$ as defined in \ref{['eq:metriclimiterror']} corresponding with approximating the metric $\alpha_n$ with \ref{['eq:metriclimit']} plotted on a log-log scale as a function of $q$ by taking $\phi=0$ when $q$ is odd, and $\phi=\pi/q$ when $q$ is even in \ref{['eq:specialmetric']}. The error scales polynomially as $\propto1/q$ (dashed line).